A055329 Number of rooted identity trees with n nodes and 4 leaves.

2, 11, 40, 109, 254, 524, 998, 1774, 2995, 4833, 7525, 11346, 16659, 23877, 33528, 46203, 62637, 83643, 110213, 143432, 184600, 235129, 296687, 371072, 460382, 566866, 693121, 841917, 1016422, 1220001, 1456473, 1729878, 2044767, 2405940

Offset: 8

Author

Christian G. Bower, May 12 2000

Keywords

• nonn
• easy

Links

• Andrew Howroyd, Table of n, a(n) for n = 8..1000
• Index entries for sequences related to rooted trees
• Index entries for linear recurrences with constant coefficients, signature (3,-1,-4,3,-1,3,0,-3,1,-3,4,1,-3,1).

Crossrefs

Column 4 of A055327.

Programs

Magma

R:=PowerSeriesRing(Integers(), 50); Coefficients(R!( x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4)) )); // G. C. Greubel, Nov 09 2023

Mathematica

Drop[CoefficientList[Series[x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4)), {x,0,50}], x], 8] (* G. C. Greubel, Nov 09 2023 *)

PARI

Vec((2 + 5*x + 9*x^2 + 8*x^3 + 5*x^4 + x^6)/((1 - x)^7*(1 + x)^3*(1 + x^2)*(1 + x + x^2)) + O(x^40)) \\ Andrew Howroyd, Aug 28 2018

SageMath

def A055329_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^3*(1-x^2)^2*(1-x^3)*(1-x^4)) ).list()
a=A055329_list(50); a[8:] # G. C. Greubel, Nov 09 2023

Formula

G.f.: x^8*(2+5*x+9*x^2+8*x^3+5*x^4+x^6)/((1-x)^7*(1+x)^3*(1+x^2)*(1+x+x^2)) - Colin Barker, Nov 07 2012

a(n) = -[n=0] + (60*n^6 -1236*n^5 +9450*n^4 -33520*n^3 +59940*n^2 -66294*n +48065)/69120 +(-1)^n*(4*n^2 -38*n +89)/512 +(3/32)*(-1)^floor((n+1)/2) + ChebyshevU(n, -1/2)/27. - G. C. Greubel, Nov 09 2023